Fuzzy number Intuitionistic fuzzy soft sets and its properties

In this paper, we have defined the notion of the fuzzy number intuitionistic fuzzy soft sets. For it, different operations such as union, intersection, complement, max, min, AND and OR have been introduced on fuzzy number intuitionistic fuzzy soft sets environment. Some examples of these operations are given and a few properties are also studied.


Introduction
The concept of an intuitionistic fuzzy set (IFS) (Attanassov, 1986 [5]) can be viewed as an alternative approach to define a fuzzy set (FS) (Zadeh, 1965) [25] in cases where available information is not sufficient for the definition of an imprecise concept by means of a conventional fuzzy set.But, Molodtsov (1999) [17] has been analyzed that these IFS and FS theories have incompatibility with the parameterization tools in dealing with the uncertainties.To handle this, Molodtsov (1999) [17] introduced the concept of a soft set as a new mathematical tool which is free from the parameterization inadequacy measures of IFS and FS.After their pioneering work, the various authors have gained their interest in it and derived various types of operators like equality, subset, superset, binary operations (Maji et al. 2003) [13].After that, many extensions of fuzzy soft sets have been proposed in different areas (Jiang et al. 2010 [10]; Maji et al. 2001 [15]; Maji et al. 2004 [14]; Majumdar and Samanta, 2010 [16]; Neoig and Sut, 2011 [19]; Zou and Xiao, 2008 [27], Alkhazaleh et al. 2011a,b,c [2-4]).Now-a-days soft sets theory has been applied in variety of disciplines such as such as smoothness of functions, game theory, operation research, measurement theory, medical diagnosis, decision making, algebra etc. (Acar, 2010 [1]; Molodtsov, 1999 [17]; Molodtsov, 2004 [18]; Roy and Maji, 2007 [20]; Celik and Yamak, 2013 [6], Majumdar and Samanta, 2010 [16], Ma et al., 2014 [12]).Yang et al. (2009) [23] presented the concept of interval-valued fuzzy soft set by combining the interval-valued fuzzy set (Gorzalczany, 1987 [8]; Zadeh 1975 [24]) and soft set (Maji et al. 2001 [15]) models.Yang et al. (2009) [23] proposed interval-valued fuzzy soft sets http://www.ispacs.com/journals/jfsva/2016/jfsva-00332/International Scientific Publications and Consulting Services (IVFSSs).Xu et al. (2010) [22] introduced vague soft set and Zhou and Li (2014) [26] defined generalized vague soft sets.Guan et al. (2013) [9] gave a new order relation of fuzzy soft set (FSSs) and Feng et al. (2014) [7] studied the decomposition of fuzzy soft sets with finite value spaces.Wang et al. (2015) [21] presented hesitant fuzzy soft sets and their corresponding aggregation operators namely hesitant fuzzy soft weighted averaging/geometric operators.Thus, motivated by the theory of intuitionistic fuzzy set and fuzzy soft set, the present paper addressed the new hybrid structure of the set theories called fuzzy number Intuitionistic fuzzy soft set.This theory combines the soft set with the fuzzy number Intuitionistic fuzzy set theory.In it, fuzzy soft set theory has been generalized by assigning a degree with the parameterization of intuitionistic fuzzy sets.Various operations, such as Union, Intersection, Compliment, Max-min, etc. have been defined and their corresponding properties in the fuzzy number Intuitionistic fuzzy soft sets environment.In order to do so, the rest of the paper has been summarized as follows.Some basic definition related to the intuitionistic fuzzy set theory and Soft set theory are given in the next section.In section 3 the notion of fuzzy number intuitionistic fuzzy soft sets is proposed.Some set theoretic operations like union, intersection, complement etc are also defined here.In Section 4 based on set theoretic operations, a number of properties of fuzzy number intuitionistic soft sets are stated and proved.In section 5 some operators such as max, min defined with their properties.The concrete conclusion about the work has been summarized in Section 6.

Preliminaries
In this section, basic concepts of intuitionistic fuzzy set and soft set have been discussed.Let X be a universe of objects and E be the set of parameters with the connection to the objects in X .[5] An intuitionistic fuzzy set A defined in X is given by }

Intuitionistic Fuzzy Set
is called the intuitionistic index or hesitancy degree of x in A .[14]:

Intuitionistic Fuzzy Soft Set
Let IFS(X) be the set of all intuitionistic fuzzy set in define the membership and the non-membership functions respectively and if will be treated as traditional fuzzy soft set.

Fuzzy Number Intuitionistic Fuzzy Soft Sets
In defining the fuzzy number intuitionistic fuzzy soft set, let X be a universal set and E be a set of parameters such that E A  .

Fuzzy Number Intuitionistic
respectively denote the degree of membership and non-membership of x to the set A , and for every , then the fuzzy number intuitionistic fuzzy soft sets set will be converted into intuitionistic fuzzy soft set and if any two values are equal in membership as well as in non-membership function, then this theory will be converted into "intervalvalued intuitionistic fuzzy soft set theory.For instance,

Null Fuzzy Number Intuitionistic Fuzzy Soft Set:
A fuzzy number intuitionistic fuzzy soft set A F, over X is said to be a null fuzzy number intuitionistic fuzzy soft set denoted by , if

Not Set:
The not set of a parameter set

Set operations on FNIFSSs
Based on the definition of fuzzy number intuitionistic fuzzy soft set theory, some basic operations like union, intersection complement etc., are define as below.FNIFSSs and its degree of membership and non-membership are defined as        is also FNIFSSs and its degree of membership and non-membership are defined as

Union of FNIFSSs
Example 3.4.In order to illustrate the complement of FNIFSS, an example as described in Example 3.1.
has been considered here.Then based on the rating values of each house for describing the "attractiveness of the house",denoted by A F, , the complement of it is given as:   A F, is a fuzzy number intuitionistic fuzzy soft subset of B G, .

b)
B G, is a fuzzy number intuitionistic fuzzy soft subset of A F, .

Properties on set operations
Here, we derive some properties on operations defined in the above section.Let B G and A F , , be the two fuzzy number intuitionistic fuzzy soft sets on universal set X , then we have the following.

Theorem 4.1. Demorgan's law
, where   and  ,  , , we have , then membership and non-membership degrees of V K, is given as Proof.II.can be proved analogously.

Max-Min Operators
In this section, we propose two operators named as max, min operators and derive some properties on fuzzy number intuitionistic fuzzy soft set theory.

Properties on Max-Min operators:
Let A F, , B G, and C H, be the fuzzy number intuitionistic fuzzy soft sets on universal set X such that , where E is the set of all parameters, then the followings theorems are defined as: Theorem 5.1.Demorgan's law , then by definition the join of these two will be denoted as O are the equivalent operators. Hence, Theorem 5.2.Associatively: ,where, ,  , , where ) , ( Similarly, we can proof the other part.

Conclusion
In this paper, the notion of fuzzy number intuitionistic fuzzy soft sets (FNIFSSs) has been proposed as a combination of soft set theory with fuzzy number intuitionistic fuzzy set.The presented theory generalizes of many theories like the fuzzy soft set theory, interval-valued intuitionistic fuzzy set theory, intuitionistic fuzzy soft set theory and interval-valued intuitionistic fuzzy soft set theory, etc. Various operations like union, intersection, compliments, max, min etc. are presented in FNIFSS environment and their corresponding properties.Numerical examples are shown to elaborate the operations and its properties.In future, we are engaged to extend this theory to some other domains.
the degree of membership and degree of non-membership of x in http://www.
is also a fuzzy number intuitionistic fuzzy soft set denoted as C the two FNIFSSs on universal set X and


Let X be the universe of discourse and E be the set of parameters.Suppose be the subset of  B G, ifand only if a) ; B A  http://www.ispacs.com/journals/jfsva/2016/jfsva-00332/International Scientific Publications and Consulting Services //www.ispacs.com/journals/jfsva/2016/jfsva-00332/International Scientific Publications and Consulting Services Since fuzzy number intuitionistic fuzzy soft sets defined on universal set X , min operator, is also a fuzzy number intuitionistic fuzzy soft set and is defined by